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Properties

Label 2-507-13.4-c1-0-15
Degree $2$
Conductor $507$
Sign $-0.455 + 0.890i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.07 − 0.623i)2-s + (−0.5 + 0.866i)3-s + (−0.222 − 0.385i)4-s − 2.80i·5-s + (1.07 − 0.623i)6-s + (4.15 − 2.40i)7-s + 3.04i·8-s + (−0.499 − 0.866i)9-s + (−1.74 + 3.02i)10-s + (1.27 + 0.733i)11-s + 0.445·12-s − 5.98·14-s + (2.42 + 1.40i)15-s + (1.45 − 2.52i)16-s + (−1.22 − 2.11i)17-s + 1.24i·18-s + ⋯
L(s)  = 1  + (−0.763 − 0.440i)2-s + (−0.288 + 0.499i)3-s + (−0.111 − 0.192i)4-s − 1.25i·5-s + (0.440 − 0.254i)6-s + (1.57 − 0.907i)7-s + 1.07i·8-s + (−0.166 − 0.288i)9-s + (−0.552 + 0.956i)10-s + (0.383 + 0.221i)11-s + 0.128·12-s − 1.60·14-s + (0.626 + 0.361i)15-s + (0.363 − 0.630i)16-s + (−0.296 − 0.513i)17-s + 0.293i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.455 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.455 + 0.890i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.455 + 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.462434 - 0.756370i\)
\(L(\frac12)\) \(\approx\) \(0.462434 - 0.756370i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (1.07 + 0.623i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 2.80iT - 5T^{2} \)
7 \( 1 + (-4.15 + 2.40i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.27 - 0.733i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.22 + 2.11i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.20 + 1.27i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.75 - 3.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.925 - 1.60i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.63iT - 31T^{2} \)
37 \( 1 + (3.94 + 2.27i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.07 + 0.623i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.19 - 2.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 12.8iT - 47T^{2} \)
53 \( 1 + 8.85T + 53T^{2} \)
59 \( 1 + (1.88 - 1.08i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.91 - 6.78i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.10 - 1.79i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.65 - 4.41i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.69iT - 73T^{2} \)
79 \( 1 + 4.02T + 79T^{2} \)
83 \( 1 + 0.652iT - 83T^{2} \)
89 \( 1 + (-5.45 - 3.14i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.68 - 5.01i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.55622995467252660959047482776, −9.720304061062422180124952557739, −8.960399786448463073445492168274, −8.251377365342590429143682578567, −7.32891268771677143288798679621, −5.51621796662655675876138743572, −4.89418352600720810456242213296, −4.12231737977480768112547845670, −1.80772737857693240126084636458, −0.77919561434232622040616540013, 1.69287772818594812639172215163, 3.14029745802830822169776279500, 4.65421019143111413033426139908, 5.98983631451222263848092118909, 6.79089610321872195729623307333, 7.73377883543380608006253012585, 8.314012325792790426973697939433, 9.160001711813818408402145506756, 10.42017986540092476061305044407, 11.10947093088256157802709170566

Graph of the $Z$-function along the critical line